
theorem
  for A, B, C being category st A, B are_dual & B, C are_equivalent
  holds A, C are_dual
proof
  let A, B, C be category such that
A1: A, B opp are_equivalent and
A2: B, C are_equivalent;
A3: B, B opp are_opposite by Def4;
  C, C opp are_opposite by Def4;
  then B opp, C opp are_equivalent by A2,A3,Th25;
  hence A, C opp are_equivalent by A1,Th4;
end;
